CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Separate Parareal iteration error from fine-method error

Statement

Let $y : [0, T] \to R^{d}$ solve a first-order initial-value problem as in First-order systems, initial value problems, and solutions on intervals, and let $x_{n}$ be the sequential fine trajectory from Fine and coarse propagators on a common macro grid on a grid $0 = t_{0} < \dots < t_{N} = T$ . Suppose the inexact Parareal ledger gives $∥ V_{n}^{K} - x_{n} ∥ \leq P_{n}^{K} + R_{n}^{K}$ , and suppose a separately established fine-method estimate gives $∥ x_{n} - y (t_{n}) ∥ \leq δ_{F}$ for some $δ_{F} \geq 0$ and every node. Then $∥ V_{n}^{K} - y (t_{n}) ∥ \leq P_{n}^{K} + R_{n}^{K} + δ_{F} .$ The quantity $δ_{F}$ must come from an independent analysis of the selected fine method and its hypotheses; this corollary does not prove a local defect, stability, or continuous-time error estimate.

Facts & Assumptions

Given: A continuous reference solution, the sequential fine trajectory, the Parareal update-defect ledger, and a separate uniform bound for the fine trajectory.

[F1]

First-order systems, initial value problems, and solutions on intervals: an ODE solution is defined on an interval, stays in its open time-state domain, and satisfies the differential equation there.

[F2]

Fine and coarse propagators on a common macro grid: the sequential fine trajectory is $x_{n + 1} = F_{n} (x_{n})$ from $x_{0}$ .

[F3]

An additive error ledger for inexact Parareal sweeps: $∥ V_{n}^{K} - x_{n} ∥ \leq P_{n}^{K} + R_{n}^{K}$ .

[A1]

The separate fine-method estimate is $∥ x_{n} - y (t_{n}) ∥ \leq δ_{F}$ for every grid node.

Proof

1.1F1F2A1F3algebra

By [F2], $x_{n}$ is the sequential fine node; since $t_{n} \in [0, T]$ , [F1] gives the solution value $y (t_{n})$ . Insert and subtract $x_{n}$ and apply the norm triangle inequality: $∥ V_{n}^{K} - y (t_{n}) ∥ \leq ∥ V_{n}^{K} - x_{n} ∥ + ∥ x_{n} - y (t_{n}) ∥$ .

2.1givenA1F3step 1.1algebra∎

Substitute the two given bounds from [A1] and [F3]. This gives the displayed conclusion, while leaving $δ_{F}$ as a separate fine-method obligation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Martin J. Gander, 50 Years of Time Parallel Time Integration (standard reference, not scraped)
Martin J. Gander, Time Parallel Time Integration, Chapter 2: Methods Based on Multiple Shooting (standard reference, not scraped)